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Secrets of image denoising cuisine. (English) Zbl 1260.94016

Summary: Digital images are matrices of equally spaced pixels, each containing a photon count. This photon count is a stochastic process due to the quantum nature of light. It follows that all images are noisy. Ever since digital images have existed, numerical methods have been proposed to improve the signal-to-noise ratio. Such ‘denoising’ methods require a noise model and an image model. It is relatively easy to obtain a noise model. As will be explained in the present paper, it is even possible to estimate it from a single noisy image.
Obtaining a convincing statistical image model is quite another story. Images reflect the world and are just as complex. Thus, any progress in image denoising implies progress in our understanding of image statistics. The present paper contains an analysis of nine recent state-of-the-art methods. This analysis shows that we are probably close to understanding digital images at a ‘patch’ scale. Recent denoising methods use thorough non-parametric estimation processes for \(8\times 8\) patches, and obtain surprisingly good denoising results.
The mathematical and experimental evidence of two recent articles suggests that we might even be close to optimal performance in image denoising. This suspicion is supported by a remarkable convergence of all analysed methods. They certainly converge in performance. We intend to demonstrate that, under different formalisms, their image models are almost equivalent. Working in the 64-dimensional ‘patch space’, all recent methods estimate local ‘sparse models’ and restore a noisy patch by finding its likeliest interpretation, given the noiseless patches.
The story will be told in an ordered fashion. Denoising methods are complex and have several indispensable ingredients. Noise model and noise estimation methods will be explained first. The four main image models used for denoising are the Markovian-Bayesian paradigm, linear transform thresholding, so-called image sparsity, and an image self-similarity hypothesis, which will also be presented. The performance of all methods depends on three generic tools: colour transform, aggregation, and an ‘oracle’ step. Their recipes will also be given. These preparations will permit us to present, in a unified terminology, the complete recipes of nine different state-of-the-art patch-based denoising methods. Three quality assessment recipes for denoising methods will also be proposed and applied to compare all methods. The paper presents an ephemeral state of the art in a rapidly changing subject, but many of the presented recipes will remain useful. Most denoising recipes can be tested directly on any digital image in the web journal Image Processing On Line.

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
65J22 Numerical solution to inverse problems in abstract spaces
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